Trigonometry Examples

$\frac{\sec (x) - \csc (x)}{\sec (x) + \csc (x)} = \frac{\tan (x) - 1}{\tan (x) + 1}$

Step 1

Start on the left side.

$\frac{\sec (x) - \csc (x)}{\sec (x) + \csc (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to $\sec (x)$ .

$\frac{\frac{1}{\cos (x)} - \csc (x)}{\sec (x) + \csc (x)}$

Step 2.2

Apply the reciprocal identity to $\csc (x)$ .

$\frac{\frac{1}{\cos (x)} - \frac{1}{\sin (x)}}{\sec (x) + \csc (x)}$

Step 2.3

Apply the reciprocal identity to $\sec (x)$ .

$\frac{\frac{1}{\cos (x)} - \frac{1}{\sin (x)}}{\frac{1}{\cos (x)} + \csc (x)}$

Step 2.4

Apply the reciprocal identity to $\csc (x)$ .

$\frac{\frac{1}{\cos (x)} - \frac{1}{\sin (x)}}{\frac{1}{\cos (x)} + \frac{1}{\sin (x)}}$

Step 3

Simplify.

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Step 3.1

Multiply the numerator and denominator of the fraction by $\cos (x) \sin (x)$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{\cos (x)} - \frac{1}{\sin (x)}}{\frac{1}{\cos (x)} + \frac{1}{\sin (x)}}$ by $\frac{\cos (x) \sin (x)}{\cos (x) \sin (x)}$ .

$\frac{\cos (x) \sin (x)}{\cos (x) \sin (x)} \cdot \frac{\frac{1}{\cos (x)} - \frac{1}{\sin (x)}}{\frac{1}{\cos (x)} + \frac{1}{\sin (x)}}$

Step 3.1.2

Combine.

$\frac{\cos (x) \sin (x) (\frac{1}{\cos (x)} - \frac{1}{\sin (x)})}{\cos (x) \sin (x) (\frac{1}{\cos (x)} + \frac{1}{\sin (x)})}$

Step 3.2

Apply the distributive property.

$\frac{\cos (x) \sin (x) \frac{1}{\cos (x)} + \cos (x) \sin (x) (- \frac{1}{\sin (x)})}{\cos (x) \sin (x) \frac{1}{\cos (x)} + \cos (x) \sin (x) \frac{1}{\sin (x)}}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $\cos (x)$ .

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Step 3.3.1.1

Factor $\cos (x)$ out of $\cos (x) \sin (x)$ .

$\frac{\cos (x) (\sin (x)) \frac{1}{\cos (x)} + \cos (x) \sin (x) (- \frac{1}{\sin (x)})}{\cos (x) \sin (x) \frac{1}{\cos (x)} + \cos (x) \sin (x) \frac{1}{\sin (x)}}$

Step 3.3.1.2

Cancel the common factor.

$\frac{\cos (x) \sin (x) \frac{1}{\cos (x)} + \cos (x) \sin (x) (- \frac{1}{\sin (x)})}{\cos (x) \sin (x) \frac{1}{\cos (x)} + \cos (x) \sin (x) \frac{1}{\sin (x)}}$

Step 3.3.1.3

Rewrite the expression.

$\frac{\sin (x) + \cos (x) \sin (x) (- \frac{1}{\sin (x)})}{\cos (x) \sin (x) \frac{1}{\cos (x)} + \cos (x) \sin (x) \frac{1}{\sin (x)}}$

Step 3.3.2

Cancel the common factor of $\sin (x)$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{\sin (x)}$ into the numerator.

$\frac{\sin (x) + \cos (x) \sin (x) \frac{- 1}{\sin (x)}}{\cos (x) \sin (x) \frac{1}{\cos (x)} + \cos (x) \sin (x) \frac{1}{\sin (x)}}$

Step 3.3.2.2

Cancel the common factor.

$\frac{\sin (x) + \sin (x) \cos (x) \frac{- 1}{\sin (x)}}{\cos (x) \sin (x) \frac{1}{\cos (x)} + \cos (x) \sin (x) \frac{1}{\sin (x)}}$

Step 3.3.2.3

Rewrite the expression.

$\frac{\sin (x) + \cos (x) \cdot - 1}{\cos (x) \sin (x) \frac{1}{\cos (x)} + \cos (x) \sin (x) \frac{1}{\sin (x)}}$

Step 3.3.3

Cancel the common factor of $\cos (x)$ .

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Step 3.3.3.1

Factor $\cos (x)$ out of $\cos (x) \sin (x)$ .

$\frac{\sin (x) + \cos (x) \cdot - 1}{\cos (x) (\sin (x)) \frac{1}{\cos (x)} + \cos (x) \sin (x) \frac{1}{\sin (x)}}$

Step 3.3.3.2

Cancel the common factor.

$\frac{\sin (x) + \cos (x) \cdot - 1}{\cos (x) \sin (x) \frac{1}{\cos (x)} + \cos (x) \sin (x) \frac{1}{\sin (x)}}$

Step 3.3.3.3

Rewrite the expression.

$\frac{\sin (x) + \cos (x) \cdot - 1}{\sin (x) + \cos (x) \sin (x) \frac{1}{\sin (x)}}$

Step 3.3.4

Cancel the common factor of $\sin (x)$ .

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Step 3.3.4.1

Cancel the common factor.

$\frac{\sin (x) + \cos (x) \cdot - 1}{\sin (x) + \sin (x) \cos (x) \frac{1}{\sin (x)}}$

Step 3.3.4.2

Rewrite the expression.

$\frac{\sin (x) + \cos (x) \cdot - 1}{\sin (x) + \cos (x)}$

Step 3.4

Simplify the numerator.

$\frac{\sin (x) - \cos (x)}{\sin (x) + \cos (x)}$

Step 4

Reorder terms.

$\frac{\sin (x) - \cos (x)}{\cos (x) + \sin (x)}$

Step 5

Rewrite $\frac{\sin (x) - \cos (x)}{\cos (x) + \sin (x)}$ as $\frac{\tan (x) - 1}{\tan (x) + 1}$ .

$\frac{\tan (x) - 1}{\tan (x) + 1}$

Step 6

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\sec (x) - \csc (x)}{\sec (x) + \csc (x)} = \frac{\tan (x) - 1}{\tan (x) + 1}$ is an identity